Two of the trilemma imply the
negation of the third, by the Intermediate Value Theorem:

If f is continuous and never
zero, then it does not change sign;

If f is never zero but changes
sign, then it is discontinuous;

If f changes sign and is
continuous, then it is somewhere zero.

Let each Stooge affirm all terms of
each of these implications, to make a troika supporting the trilemma.

A real continuous nonzero
sign-changing function is a curious beast; it’s a kind of continuous Heap,
changing sides without crossing the boundary. Therefore let us call such a
function a “Smuggler”, and this the Smuggler Trilemma.

Consider the tree. First it is less
than ten million microns tall, later it is more than ten million microns tall;
it grows continuously; yet you cannot find a microsecond in which it is exactly ten million microns tall! So let
us call this the Paradox of the Tree.

Now consider this Unmean Values Trilemma:

* f(x) is differentiable on the
interval [a,b];

* f(a) < f(b) ;

* df/dx is positive nowhere on
[a,b].

You can also call this the Still Growth Trilemma. Any two of the
trilemma implies the negation of the third, by the Mean Values Theorem. As
ever, this unpacks into three deduction rules, and expands into a troika.

Consider the tree. It increases
in size, at a continuous growth rate of zero. That is the Second Paradox of the Tree.

The middle term
says that P is ‘inductive’; which means an infinite conjunction of
implications: (P(1) implies P(2)) and
(P(2) implies P(3)) and (P(3) implies P(4)) and …Therefore mathematical induction is a sorites
of infinite length.

Denying the
conclusion creates a trilemma:

1 has property P;

For
any positive integer n, if n has property P then so does n+1;

Some
positive integer N does not have property P.

This is the Disinduction Trilemma. It is also known
as the Paradox of the Heap: for let
P = ‘does not make a heap of sand grains’. Surely 1 grain of sand is not a
heap; and surely adding one single grain of sand to a non-heap will not make it
a heap, so non-heapness is inductive; yet surely there is some number N,
denumerating a sand heap!

Disinduction can be
expanded into this troika:

Moe:
1 has property P, P is inductive, and every N has property P.

Larry:
P is inductive, some N does not have property P, and neither does 1.

Curly:
1 has property P, some N does not, and P is not inductive.

It also can be
unpacked into these three deduction rules:

If 1
has property P, and P is inductive, then every N has property P.

IfP is inductive, and some N does not have
property P, then neither does 1.

Wednesday, June 26, 2013

An anti-sorites is a multilemma; that is, a set of
statements, not all of which can be true. Therefore all but one true implies
the last is false; a form of sorites reasoning. Any anti-sorites encodes
several sorites at once.

You make an
anti-sorites by appending the negation of a sorite’s classical conclusion. This
you can then ‘untangle’, by row swaps and modal identities. For instance, take
this sorites:

Some
lunches are nutritious;Anything
nutritious is good for you;Only
valuable things are good for you;Anything
valuable is paid for;free
means paid for.

The
logical conclusion to this sorites is Some
lunch is free.
- being nutritious, good for you, valuable, paid for, and hence free. After
all, you paid for it.
To make an anti-sorites, replace that conclusion with “no lunch is free”,and get:

Some lunches are nutritious;Anything
nutritious is good for you;Only
valuable things are good for you;Anything
valuable is paid for;Free
means paid for;No
lunch is free.

This anti-sorites, untangled
by row swaps and modal identities, becomes;

Some lunches are
nutritious;
Anything nutritious is good for you;
Anything good for you is valuable;
Anything valuable is paid for;
Anything paid for is free;
No lunch is free.

This is a SAAAAN
anti-sorites. The inner “all” sequence collapses to “Anything nutritious is
free”, resulting in a Some-All-None trilemma. Either the initial “some”
statement is false, or the final “none” statement is false, or one of the chain
of “all”s fails.Here’s an anti-sorites,
row-swapped and then remodulated:

Butterflies are free;Not
all lunches are bland;Only
butterflies are beautiful;All
unbeautiful things are bland;There’s
no such thing as a free lunch.

Not
all lunches are bland;All
unbeautiful things are bland;Only
butterflies are beautiful;Butterflies
are free;There’s
no such thing as a free lunch.

Some
lunches are not bland;All
not-bland things are beautiful;All
beautiful things are butterflies;All
butterflies are free;No
lunch is free.

Here are some
anti-sorites derived from sorites by Lewis Carroll, then untangled:

No interesting poems
are unpopular among people of real taste;No
modern poetry is free of affectation;All
of your poems are on the subject of soap-bubbles;No
affected poetry is popular among people of real taste;No
ancient poem is on the subject of soap-bubbles;Some
of your poems are interesting.

Some
interesting poems are yours;All
of your poems are on the subject of soap-bubbles;All
poems on the subject of soap-bubbles are modern;All
modern poetry is affected;All
affected poetry is unpopular among people of real taste;No
interesting poems are unpopular among people of real taste.

No
kitten that loves fish is unteachable;No
kitten without a tail will play with a gorilla;Kittens
with whiskers always love fish;No
teachable kitten has green eyes;No
kittens have tails unless they have whiskers;Some
kitten with green eyes will play with a gorilla.

Some
kitten with green eyes will play with a gorilla;Any
kitten that will play with a gorilla has a tail;All
kittens with tails have whiskers;All
kittens with whiskers love fish;All
kittens who love fish are teachable;No
kittens with green eyes are teachable.

Things
sold on the street are of no great value;Nothing
but rubbish can be had for a song;Eggs
of the Great Auk are very valuable;It
is only what is sold on the street that is really rubbish;Eggs
of the Great Auk can be had for a song.

Eggs
of the Great Auk can be had for a song;Anything
that can be had for a song is really rubbish;Anything
that is really rubbish is sold on the street;Anything
sold on the street is not very valuable;Eggs
of the Great Auk are very valuable.

This
multilemma has a single object at beginning and end, with a property inverting
during the A sequence; so an “xAAA~x” anti-sorites.We
know that one of the statements in the Great Auk Anti-Sorites is false. This is
a lot more diffuse than a trilemma. An anti-sorites is like the game of
telephone, with an error happening somewhere in the deductive chain; then a ‘some’
vanishes into a ‘none’, or the Great Auk’s eggs turn upside down.

All of these can
be supported by troikas. Each Stooge need merely deny one of three different
terms of the anti-sorites. Perhaps Moe could deny the Some statement, Larry
could deny one of the Alls, and Curly deny the None.

If Moe gives time a beginning but
no end, then the trilemma is: time is finite; past time is unbounded; past time
is linear. If Moe gives time an end but no beginning, then the trilemma is:
time is finite; future time is unbounded; future time is linear.

These
are both what I call a Line/Ray/Loop
Troika. The line, ray and loop needn’t be in time; they can be any ordered
sequence, continuous or discrete. In a line/ray/loop troika:

Moe sees a Ray: the sequence has
an endpoint.

Larry sees a Line: the sequence
extends to infinity.

Curly sees a Loop: the sequence
cycles.

Majorities affirm this “Finite Boundless Line” trilemma:

The sequence is finite;

The sequence is unbounded;

The sequence is linear;

-but
all agree that the sequence is not all three!

Now consider this “Agrippa Troika”:

Moe:
There is an unexplained first explanation.

Larry:
Explanations regress to infinity.

Curly:
Explanation is circular.

In this troika, dogmatic Moe has
an explanatory Ray, infinite-regressing Larry has an explanatory Line, and
circular-reasoning Curly has an explanatory Loop. Majorities affirm:

Explanation is finite;

Explanation
is complete;

Explanation
is noncircular.

But not all three! That is the “Munchausen Trilemma”:

Any explanation can be at most two of:

Finite; that is, some explanation
explains all others;

Complete;
that is, every explanation is explained;

Noncircular,
that is, there are no explanatory loops.

The Munchausen Trilemma describes
a finite boundless line of explanations. Finite, complete and noncircular is
called normality; an illusion created by paradox.

The line/ray/loop troika applies
to the Paradox of the First Cause. Define the First Cause as the cause of all
causes; then the Paradox of the First Cause is; what caused the first cause? If
there be any such cause, then let us call it a zeroth cause. Is there a zeroth
cause?

There are three possibilities:

In the Line, there is no first
cause, but instead an endless backwards sequence of causation.The Line is a theory of infinitely deep
causation.

In the Ray, there is a first
cause but no zeroth cause; so the first cause is uncaused. The Ray is a theory
of chaotic causation.

And in the Loop, there is a first
cause and a zeroth cause; and these cause each other. The Loop is a theory of
circular causation.

The Line corresponds to chaos
theory, with infinite fractal complexity. The Ray corresponds to quantum
theory, based upon intrinsic random chance. The Loop corresponds to Goedelian
metamath, with self-referential paradox. Chaos, quantum and metamath are the 20th
century’s three-fold retort to determinism. Of the three:

In Ray and Loop, causation has
finite depth;

In Loop and Line, causation is
recursive;

In Line and Ray, causation does
not loop.

So
by 2/3 majorities: causation is finite; causation is recursive; causation is
linear; but not all three! That’s the Causation Trilemma, a finite boundless
line. Cause has a beginning, every cause has a cause, cause is one-way; deny
one!

Dual to the First Cause is the
Final Effect; the effect of all effects. Does the Final Effect have any effect?
Either there is no Final Effect, or the Final Effectis ineffectual, or there is a Postfinal
Effect.

This
translates, as before, into Line, Ray and Loop. In the Line, effect is linear,
recursive and infinite; there is no final effect. In the Ray, effect is linear,
nonrecursive, and finite; the final effect has no effect. In the Loop, effect
is circular, recursive and finite; postfinal and final are effects of each
other.

OfLine, Ray and Loop, by 2/3 majority each:
effect is linear; effect is finite; effect is recursive: but never all three.
That’s the Effectuation Trilemma, another finite boundless line.Effect has an end, every effect has an
effect, effect is one-way; deny one!